View source for Stability for a part of the variables

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[[Lyapunov stability|Lyapunov stability]] of the solution  $  x = 0 $
relative not to all but only to certain variables  $  x _ {1} \dots x _ {k} $, 
$  k < n $, 
of a system of ordinary differential equations

$$ \tag{1 }
\dot{x} _ {s}  =  X _ {s} ( t, x _ {1} \dots x _ {n} ),\ \ 
s = 1 \dots n.
$$

Here  $  X _ {s} ( t, x) $
are given real-valued continuous functions, satisfying in the domain

$$ \tag{2 }
t  \geq   0,\ \ 
\sum _ {i = 1 } ^ { k } 
x _ {i}  ^ {2}  \leq   \textrm{ const } ,\ \ 
\sum _ {j = k + 1 } ^ { n } 
x _ {j}  ^ {2}  <  \infty
$$

the conditions for the existence and uniqueness of the solution  $  x ( t;  t _ {0} , x _ {0} ) $; 
moreover,

$$ 
X _ {s} ( t, 0)  \equiv  0,\  s = 1 \dots n,
$$

and any solution is defined for all  $  t \geq  t _ {0} \geq  0 $
for which  $  \sum _ {i = 1 }   ^ {k} x _ {i}  ^ {2} \leq  H $.

Put  $  x _ {i} = y _ {i} $
for  $  i = 1 \dots k $; 
$  x _ {k + j }  = z _ {j} $
for  $  j = 1 \dots m $, 
$  n = k + m $
and  $  m \geq  0 $; 
let

$$ 
\| y \|  = \ 
\left ( \sum _ {i = 1 } ^ { k }  y _ {i}  ^ {2} \right )  ^ {1/2} ,\ \ 
\| z \|  = \ 
\left ( \sum _ {j = 1 } ^ { m }  z _ {j}  ^ {2} \right )  ^ {1/2} ,
$$

$$ 
\| x \|  =  \left ( \sum _ {s = 1 } ^ { n }  x _ {s}  ^ {2} \right )  ^ {1/2} .
$$

The solution  $  x = 0 $
of the system (1) is called: a) stable relative to  $  x _ {1} \dots x _ {k} $
or  $  y $-
stable if

$$ 
( \forall \epsilon > 0)
( \forall t _ {0} \in I)
( \exists \delta > 0)
( \forall x _ {0} \in B _  \delta  )
( \forall t \in J  ^ {+} ):
$$

$$ 
\| y ( t;  t _ {0} , x _ {0} ) \|  <  \epsilon ,
$$

i.e. for any given numbers  $  \epsilon > 0 $(
$  \epsilon < H $) 
and  $  t _ {0} \geq  0 $
one can find a number  $  \delta ( \epsilon , t _ {0} ) > 0 $
such that for every perturbation  $  x _ {0} $
satisfying the condition  $  \| x _ {0} \| \leq  \delta $
and for every  $  t > t _ {0} $
the solution  $  y ( t;  t _ {0} , x _ {0} ) $
satisfies the condition  $  \| y \| < \epsilon $;

b)  $  y $-
unstable in the opposite case, i.e. if

$$ 
( \exists \epsilon > 0)
( \exists t _ {0} \in I)
( \forall \delta > 0)
( \exists x _ {0} \in B _  \delta  )
( \exists t \in J  ^ {+} ):
$$

$$ 
\| y ( t;  t _ {0} , x _ {0} ) \|  \geq   \epsilon ;
$$

c)  $  y $-
stable uniformly in  $  t _ {0} $
if in definition a) for every  $  \epsilon > 0 $
the number  $  \delta ( \epsilon ) $
may be chosen independently of  $  t _ {0} $;

d) asymptotically  $  y $-
stable if it is  $  y $-
stable and if for every  $  t _ {0} \geq  0 $
there exists a  $  \delta _ {1} ( t _ {0} ) > 0 $
such that

$$ 
\lim\limits _ {t \rightarrow \infty }  \| y ( t;  t _ {0} , x _ {0} ) \|  =  0 \ \ 
\textrm{ for }  \| x _ {0} \| \leq  \delta _ {1} .
$$

Here  $  I = [ 0, \infty ) $, 
$  J  ^ {+} $
is the maximal right interval on which  $  x ( t;  t _ {0} , x _ {0} ) $
is defined,  $  B _  \delta  = \{ {x \in \mathbf R  ^ {n} } : {\| x \| < \delta } \} $; 
in case d), besides the conditions stated above it is assumed that all solutions of the system (1) exist on  $  [ t _ {0} , \infty ) $.

The statement of the problem of stability for a part of the variables was given by A.M. Lyapunov [[#References|[1]]] as a generalization of the stability problem with respect to all variables  $  ( k = n) $. 
For a solution of this problem it is particularly effective to apply the method of Lyapunov functions, suitably modified (cf. [[#References|[2]]], and [[Lyapunov function|Lyapunov function]]) for the problem of  $  y $-
stability. At the basis of this method there are a number of theorems generalizing the classical theorem of Lyapunov.

Consider a real-valued function  $  V ( t, x) \in C  ^ {1} $, 
$  V ( t, 0) = 0 $, 
and at the same time its total derivative with respect to time, using (1):

$$ 
\dot{V}  = \ 

\frac{\partial  V }{\partial  t }
 +
\sum _ {s = 1 } ^ { n } 

\frac{\partial  V }{\partial  x _ {s} }

X _ {s} .
$$

A function  $  V ( t, x) $
of constant sign is called  $  y $-
sign-definite if there exists a positive-definite function  $  W ( y) $
such that in the region (2),

$$ 
V ( t, x)  \geq   W ( y) \ \ 
\textrm{ or } \ \ 
- V ( t, x)  \geq   W ( y).
$$

A bounded function  $  V ( t, x) $
is said to admit an infinitesimal upper bound for  $  x _ {1} \dots x _ {p} $
if for every  $  l > 0 $
there exists a  $  \lambda ( l) $
such that

$$ 
| V ( t, x) |  <  l
$$

for  $  t \geq  0 $, 
$  \sum _ {i = 1 }   ^ {p} x _ {i}  ^ {2} < \lambda $, 
$  - \infty < x _ {p + 1 }  \dots x _ {n} < \infty $.

===Theorem 1.===
If the system (1) is such that there exists a  $  y $-
positive-definite function  $  V ( t, x) $
with derivative  $  \dot{V} \leq  0 $, 
then the solution  $  x = 0 $
is  $  y $-
stable.

===Theorem 2.===
If the conditions of theorem 1 are fulfilled and if, moreover,  $  V $
admits an infinitesimal upper bound for  $  x $, 
then the solution  $  x = 0 $
of the system (1) is  $  y $-
stable uniformly in  $  t _ {0} $.

===Theorem 3.===
If the conditions of theorem 1 are fulfilled and if, moreover,  $  V $
admits an infinitesimal upper bound for  $  y $, 
then for any  $  \epsilon > 0 $
one can find a  $  \delta _ {2} ( \epsilon ) > 0 $
such that  $  t _ {0} \geq  0 $, 
$  \| y _ {0} \| \leq  \delta _ {2} $, 
$  0 \leq  \| z _ {0} \| < \infty $
implies the inequality

$$ 
\| y ( t;  t _ {0} , x _ {0} ) \|  < \ 
\epsilon \  \textrm{ for }  \textrm{ all }  t \geq  t _ {0} .
$$

===Theorem 4.===
If the system (1) is such that there exists a  $  y $-
positive-definite function  $  V $
admitting an infinitesimal upper bound for  $  x _ {1} \dots x _ {p} $(
$  k \leq  p \leq  n $) 
and with negative-definite derivative  $  \dot{V} $
for  $  x _ {1} \dots x _ {p} $, 
then the solution  $  x = 0 $
of the system (1) is asymptotically  $  y $-
stable.

For the study of  $  y $-
instability, Chetaev's instability theorem (cf. [[Chetaev function|Chetaev function]]) has been successfully applied, as well as certain other theorems. Conditions for the converse of a number of theorems on  $  y $-
stability have been established; for example, the converses of theorems 1, 2 as well as of theorem 4 for  $  p = k $. 
Methods of differential inequalities and Lyapunov vector functions have been applied to establish theorems on asymptotic  $  y $-
stability in the large, on first-order approximations, etc. (cf. [[#References|[3]]], ).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,   ''Mat. Sb.'' , '''17''' :  2  (1893)  pp. 253–333</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Rumyantsev,   "On stability of motion for a part of the variables"  ''Vestn. Moskov. Univ. Ser. Mat. Mekh. Astron. Fiz. Khim.'' :  4  (1957)  pp. 9–16  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.S. Oziraner,   V.V. Rumyantsev,   "The method of Lyapunov functions in the stability problem for motion with respect to a part of the variables"  ''J. Appl. Math. Mech.'' , '''36'''  (1972)  pp. 341–362  ''Prikl. Mat. i Mekh.'' , '''36''' :  2  (1972)  pp. 364–384</TD></TR></table>

====Comments====
Stability for a part of the variables is also called partial stability and occasionally conditional stability, [[#References|[a1]]]. However, the latter phrase is also used in a different meaning: Let  $  C $
be a class of trajectories,  $  x ( t ;  t _ {0} , x _ {0} ) $
a trajectory in  $  C $. 
This trajectory is stable relative to  $  C $
if for a given  $  \epsilon > 0 $
there exists a  $  \delta > 0 $
such that for each trajectory  $  \widetilde{x}  ( t ;  t _ {0} , \widetilde{x}  {} _ {0} ) $
one has that  $  \| x _ {0} - \widetilde{x}  {} _ {0} \| \leq  \delta $
implies  $  \| x( t ;  t _ {0} , x _ {0} ) - \widetilde{x}  ( t ;  t _ {0} , \widetilde{x}  {} _ {0} ) \| \leq  \epsilon $. 
If  $  C $
is not the class of all trajectories, such a  $  x ( t;  t _ {0} , x _ {0} ) $
is called conditionally stable, [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Hahn,   "Stability of motion" , Springer  (1965)  pp. §55</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Lefshetz,   "Differential equations: geometric theory" , Dover, reprint  (1977)  pp. 78, 83</TD></TR></table>